Assignment 2

Question 1 (Exercise 3.8)

Let \(C\) and \(D\) be any events for which \(\mathbb{P}(C)>0\) and \(\mathbb{P}(D)>0\). Prove that, if \(\mathbb{P}(C \mid D)> \mathbb{P}(C)\), then \(\mathbb{P}(D \mid C) > \mathbb{P}(D)\).

Question 2 (Exercise 3.26)

At a party you hear that Leo’s birthday is in an earlier month of the year than Matt’s (denote this event by ‘\(L < M\)’). Then you meet Keanu and wonder “what is the chance that Keanu’s birthday is in an earlier month than Leo’s?”. Suppose that each of the three is equally likely to be born in any month of the year and calculate:

  1. \(\pr{L < M}\); (hint: Use P4, with the partition of events \(L_j\) = Leo’s birthday is in month \(j\), with \(j=1,2,\dots, 12\).)

  2. \(\cpr{L_j}{L < M}\), for \(j = 1,2,\dots,12\);

  3. \(\cpr{K < L}{L < M}\).

hint: For the final part of the question, use the conditional version of the partition theorem i.e., \[ \cpr{A}{C} = \sum_j \pr{A \mid B_j\cap C}\cpr{B_j}{C} \] where the \(B_j\) form a partition; also use that conditionally on \(L_j\), \(K < L\) is independent of \(L<M\).

Question 3 (comparative judgement question)

Suppose you are trying to explain the difference between independence and conditional independence to someone else. Suggest an example of two events which are conditionally independent given a third event, but not independent themselves. Your explanation should be around 1-2 paragraphs.

Next week, you will be asked to look at pairs of these summaries, submitted by your peers, and select the summary which you think best encompasses this idea.